Bad news: there are fundamental conflicts between blockchain, at least in its original form, and data privacy regulations like GDPR. Good news: these conflicts provide a good opportunity to learn about what is actually contained in both.
The core property of blockchain that is novel and has been important for non-scam applications is immutability. If you have a network that is working properly, a bit of information you put on the blockchain will stay there unchanged. You will not be able to take it back, and no one else will be able to tamper with it either. This was important for the prototypical blockchain application, cryptocurrency, as it was necessary to hold people’s feet to the fire regarding their transactions - if I can change or retract what I put on the blockchain, I can undo my spending after I have run off with the goods and then double-spend my coins.
A few newer brands of blockchain (EOS comes to mind) have methods for retracting transactions, but these are controversial. For human and technical reasons, take-backsies on blockchain transactions may always range from slow and difficult to impossible.
On the GDPR side, a key privacy provision is the right to be forgotten - you can write to Google or Facebook and tell them to delete all the information they have about you. This clause has proven influential and has been incorporated into newer laws like California’s CCPA, and it will likely be imitated again in the future. From a concrete information technology standpoint, forgetting someone means going into your database to delete all the records related to them and their relationship with your business. A blockchain is really just an immutable database, so if you are using a blockchain in your technology stack right to be forgotten vs. immutability is a real headache. Immutability is explicitly a “real pain-in-the-ass to delete things” property, and this is a problem if you are legally required to delete things on request.
There are some things that can be done to mitigate these problems but at the end of the day it is actually the essential, novel property of blockchains that is the problem.